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Simultaneous Elements of Prescribed Multiplicative Orders 2
Simultaneous Elements Of Prescribed Multiplicative Orders N. A. Carella Abstract: Let u 6= ±1, and v 6= ±1 be a pair of fixed relatively prime squarefree integers, and let d ≥ 1, and e ≥ 1 be a pair of fixed integers. It is shown that there are infinitely many primes p ≥ 2 such that u and v have simultaneous prescribed multiplicative orders ordp u = (p − 1)/d and ordp v = (p − 1)/e respectively, unconditionally. In particular, a squarefree odd integer u > 2 and v = 2 are simultaneous primitive roots and quadratic residues (or quadratic nonresidues) modulo p for infinitely many primes p, unconditionally. Contents 1 Introduction 1 2 Divisors Free Characteristic Function 4 3 Finite Summation Kernels 5 4 Gaussian Sums, And Weil Sums 7 5 Incomplete And Complete Exponential Sums 7 6 Explicit Exponential Sums 11 7 Equivalent Exponential Sums 12 8 Double Exponential Sums 13 9 The Main Term 14 10 The Error Terms 15 arXiv:2103.04822v1 [math.GM] 5 Mar 2021 11 Simultaneous Prescribed Multiplicative Orders 16 12 Probabilistic Results For Simultaneous Orders 18 1 Introduction The earliest study of admissible integers k-tuples u1, u2, . , uk ∈ Z of simultaneous prim- itive roots modulo a prime p seems to be the conditional result in [21]. Much more general March 9, 2021 AMS MSC2020 :Primary 11A07; Secondary 11N13 Keywords: Distribution of primes; Primes in arithmetic progressions; Simultaneous primitive roots; Simul- taneous prescribed orders; Schinzel-Wojcik problem. 1 Simultaneous Elements Of Prescribed Multiplicative Orders 2 results for admissible rationals k-tuples u1, u2, . , uk ∈ Q of simultaneous elements of independent (or pseudo independent) multiplicative orders modulo a prime p ≥ 2 are considered in [26], and [14]. -
Idempotent Factorizations of Square-Free Integers
information Article Idempotent Factorizations of Square-Free Integers Barry Fagin Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-333-7377 Received: 20 June 2019; Accepted: 3 July 2019; Published: 6 July 2019 Abstract: We explore the class of positive integers n that admit idempotent factorizations n = p¯q¯ such that l(n)¶(p¯ − 1)(q¯ − 1), where l is the Carmichael lambda function. Idempotent factorizations with p¯ and q¯ prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p¯ and/or q¯. Idempotent factorizations are exactly those p¯ and q¯ that generate correctly functioning keys in the Rivest–Shamir–Adleman (RSA) 2-prime protocol with n as the modulus. While the resulting p¯ and q¯ have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution. Keywords: cryptography; abstract algebra; Rivest–Shamir–Adleman (RSA); computer science education; cryptography education; number theory; factorization MSC: [2010] 11Axx 11T71 1. Introduction Certain square-free positive integers n can be factored into two numbers (p¯, q¯) such that l(n)¶ (p¯ − 1)(q¯ − 1), where l is the Carmichael lambda function. -
The Ray Attack on RSA Cryptosystems’, in R
The ray attack, an inefficient trial to break RSA cryptosystems∗ Andreas de Vries† FH S¨udwestfalen University of Applied Sciences, Haldener Straße 182, D-58095 Hagen Abstract The basic properties of RSA cryptosystems and some classical attacks on them are described. Derived from geometric properties of the Euler functions, the Euler function rays, a new ansatz to attack RSA cryptosystemsis presented. A resulting, albeit inefficient, algorithm is given. It essentially consists of a loop with starting value determined by the Euler function ray and with step width given by a function ωe(n) being a multiple of the order ordn(e), where e denotes the public key exponent and n the RSA modulus. For n = pq and an estimate r < √pq for the smaller prime factor p, the running time is given by T (e,n,r)= O((r p)lnelnnlnr). − Contents 1 Introduction 1 2 RSA cryptosystem 2 2.1 Properties of an RSA key system . ... 4 2.2 ClassicalRSAattacks. 5 3 The Euler function ray attack 7 3.1 The ω-function and the order of a number modulo n ............... 7 3.2 Properties of composed numbers n = pq ..................... 10 3.3 Thealgorithm................................... 13 arXiv:cs/0307029v1 [cs.CR] 11 Jul 2003 4 Discussion 15 A Appendix 15 A.1 Euler’sTheorem.................................. 15 A.2 The Carmichael function and Carmichael’s Theorem . ......... 16 1 Introduction Since the revolutionary idea of asymmetric cryptosystems was born in the 1970’s, due to Diffie and Hellman [4] and Rivest, Shamir and Adleman [9], public key technology became an in- dispensable part of contemporary electronically based communication. -
Elementary Number Theory
Elementary Number Theory Peter Hackman HHH Productions November 5, 2007 ii c P Hackman, 2007. Contents Preface ix A Divisibility, Unique Factorization 1 A.I The gcd and B´ezout . 1 A.II Two Divisibility Theorems . 6 A.III Unique Factorization . 8 A.IV Residue Classes, Congruences . 11 A.V Order, Little Fermat, Euler . 20 A.VI A Brief Account of RSA . 32 B Congruences. The CRT. 35 B.I The Chinese Remainder Theorem . 35 B.II Euler’s Phi Function Revisited . 42 * B.III General CRT . 46 B.IV Application to Algebraic Congruences . 51 B.V Linear Congruences . 52 B.VI Congruences Modulo a Prime . 54 B.VII Modulo a Prime Power . 58 C Primitive Roots 67 iii iv CONTENTS C.I False Cases Excluded . 67 C.II Primitive Roots Modulo a Prime . 70 C.III Binomial Congruences . 73 C.IV Prime Powers . 78 C.V The Carmichael Exponent . 85 * C.VI Pseudorandom Sequences . 89 C.VII Discrete Logarithms . 91 * C.VIII Computing Discrete Logarithms . 92 D Quadratic Reciprocity 103 D.I The Legendre Symbol . 103 D.II The Jacobi Symbol . 114 D.III A Cryptographic Application . 119 D.IV Gauß’ Lemma . 119 D.V The “Rectangle Proof” . 123 D.VI Gerstenhaber’s Proof . 125 * D.VII Zolotareff’s Proof . 127 E Some Diophantine Problems 139 E.I Primes as Sums of Squares . 139 E.II Composite Numbers . 146 E.III Another Diophantine Problem . 152 E.IV Modular Square Roots . 156 E.V Applications . 161 F Multiplicative Functions 163 F.I Definitions and Examples . 163 CONTENTS v F.II The Dirichlet Product . -
Computing Multiplicative Order and Primitive Root in Finite Cyclic Group
Computing Multiplicative Order and Primitive Root in Finite Cyclic Group Shri Prakash Dwivedi ∗ Abstract Multiplicative order of an element a of group G is the least positive integer n such that an = e, where e is the identity element of G. If the order of an element is equal to |G|, it is called generator or primitive root. This paper describes the algorithms for computing multiplicative order and Z∗ primitive root in p, we also present a logarithmic improvement over classical algorithms. 1 Introduction Algorithms for computing multiplicative order or simply order and primitive root or generator are important in the area of random number generation and discrete logarithm problem among oth- Z∗ ers. In this paper we consider the problem of computing the order of an element over p, which is Z∗ multiplicative group modulo p . As stated above order of an element a ∈ p is the least positive integer n such that an = 1. In number theoretic language, we say that the order of a modulo m is n, if n is the smallest positive integer such that an ≡ 1( mod m). We also consider the related Z∗ Z∗ problem of computing the primitive root in p. If order of an element a ∈ n usually denoted as Z∗ ordn(a) is equal to | n| i.e. order of multiplicative group modulo n, then a is called called primi- Z∗ tive root or primitive element or generator [3] of n. It is called generator because every element Z∗ in n is some power of a. Efficient deterministic algorithms for both of the above problems are not known yet. -
Composite Numbers That Give Valid RSA Key Pairs for Any Coprime P
information Article Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p Barry Fagin ID Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-339-4514 Received: 13 August 2018; Accepted: 25 August 2018; Published: 28 August 2018 Abstract: RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific (s, r) pairs can be found for which encryption and decryption will be correct for any (e, d) exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime r - s. We give necessary and sufficient conditions for s with this property. Keywords: cryptography; abstract algebra; RSA; computer science education; cryptography education MSC: [2010] 11Axx 11T71 1. Notation and Background Consider the RSA public-key cryptosystem and its operations of encryption and decryption [1]. Let (p, q) be primes, n = p ∗ q, f(n) = (p − 1)(q − 1) denote Euler’s totient function and (e, d) the ∗ Z encryption/decryption exponent pair chosen such that ed ≡ 1. Let n = Un be the group of units f(n) Z mod n, and let a 2 Un. Encryption and decryption operations are given by: (ae)d ≡ (aed) ≡ (a1) ≡ a mod n We consider the case of RSA encryption and decryption where at least one of (p, q) is a composite number s. -
Arxiv:Math/0412262V2 [Math.NT] 8 Aug 2012 Etrgae Tgte Ihm)O Atnscnetr and Conjecture fie ‘Artin’S Number on of Cojoc Me) Domains’
ARTIN’S PRIMITIVE ROOT CONJECTURE - a survey - PIETER MOREE (with contributions by A.C. Cojocaru, W. Gajda and H. Graves) To the memory of John L. Selfridge (1927-2010) Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the lit- erature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contribu- tions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Woj- ciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean domains’. Contents 1. Introduction 2 2. Naive heuristic approach 5 3. Algebraic number theory 5 3.1. Analytic algebraic number theory 6 4. Artin’s heuristic approach 8 5. Modified heuristic approach (`ala Artin) 9 6. Hooley’s work 10 6.1. Unconditional results 12 7. Probabilistic model 13 8. The indicator function 17 arXiv:math/0412262v2 [math.NT] 8 Aug 2012 8.1. The indicator function and probabilistic models 17 8.2. The indicator function in the function field setting 18 9. Some variations of Artin’s problem 20 9.1. Elliptic Artin (by A.C. Cojocaru) 20 9.2. Even order 22 9.3. Order in a prescribed arithmetic progression 24 9.4. Divisors of second order recurrences 25 9.5. Lenstra’s work 29 9.6. -
A Quantum Primality Test with Order Finding
A quantum primality test with order finding Alvaro Donis-Vela1 and Juan Carlos Garcia-Escartin1,2, ∗ 1Universidad de Valladolid, G-FOR: Research Group on Photonics, Quantum Information and Radiation and Scattering of Electromagnetic Waves. o 2Universidad de Valladolid, Dpto. Teor´ıa de la Se˜nal e Ing. Telem´atica, Paseo Bel´en n 15, 47011 Valladolid, Spain (Dated: November 8, 2017) Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat’s theorem. For an integer N, the test tries to find an element of the multiplicative group of integers modulo N with order N − 1. If one is found, the number is known to be prime. During the test, we can also show most of the times N is composite with certainty (and a witness) or, after log log N unsuccessful attempts to find an element of order N − 1, declare it composite with high probability. The algorithm requires O((log n)2n3) operations for a number N with n bits, which can be reduced to O(log log n(log n)3n2) operations in the asymptotic limit if we use fast multiplication. Prime numbers are the fundamental entity in number For a prime N, ϕ(N) = N 1 and we recover Fermat’s theory and play a key role in many of its applications theorem. − such as cryptography. Primality tests are algorithms that If we can find an integer a for which aN−1 1 mod N, determine whether a given integer N is prime or not. -
Exponential and Character Sums with Mersenne Numbers
Exponential and character sums with Mersenne numbers William D. Banks Dept. of Mathematics, University of Missouri Columbia, MO 65211, USA [email protected] John B. Friedlander Dept. of Mathematics, University of Toronto Toronto, Ontario M5S 3G3, Canada [email protected] Moubariz Z. Garaev Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´exico C.P. 58089, Morelia, Michoac´an, M´exico [email protected] Igor E. Shparlinski Dept. of Computing, Macquarie University Sydney, NSW 2109, Australia [email protected] Dedicated to the memory of Alf van der Poorten 1 Abstract We give new bounds on sums of the form Λ(n) exp(2πiagn/m) and Λ(n)χ(gn + a), 6 6 nXN nXN where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums exp(2πiaMp/m) and χ(Mp) 6 6 pXN pXN p with Mersenne numbers Mp = 2 − 1, where p is prime. AMS Subject Classification Numbers: 11L07, 11L20. 1 Introduction Let m be an arbitrary natural number, and let a and g be integers that are coprime to m. Our aim in the present note is to give bounds on exponential sums and multiplicative character sums of the form n n Sm(a; N)= Λ(n)em(ag ) and Tm(χ, a; N)= Λ(n)χ(g + a), 6 6 nXN nXN where em is the additive character modulo m defined by em(x) = exp(2πix/m) (x ∈ R), and χ is a nontrivial multiplicative character modulo m. -
Arithmetic Properties of Φ(N)/Λ(N) and the Structure of the Multiplicative Group Modulo N
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Missouri: MOspace Arithmetic Properties of '(n)/λ(n) and the Structure of the Multiplicative Group Modulo n William D. Banks Department of Mathematics, University of Missouri Columbia, MO 65211 USA [email protected] Florian Luca Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´exico C.P. 58089, Morelia, Michoac´an, M´exico [email protected] Igor E. Shparlinski Department of Computing Macquarie University Sydney, NSW 2109, Australia [email protected] Abstract For a positive integer n, we let '(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio '(n)/λ(n) and study various arithmetic properties of ξ(n). AMS-classification: 11A25 Keywords: Euler function, Carmichael function. 1 1 Introduction and Notation Let '(n) denote the Euler function, which is defined as usual by '(n) = #(Z=nZ)× = pν−1(p − 1); n ≥ 1: ν pYk n The Carmichael function λ(n) is defined for all n ≥ 1 as the largest order of any element in the multiplicative group (Z=nZ)×. More explicitly, for any prime power pν, one has pν−1(p − 1) if p ≥ 3 or ν ≤ 2; λ(pν) = 2ν−2 if p = 2 and ν ≥ 3; and for an arbitrary integer n ≥ 2, ν1 νk λ(n) = lcm λ(p1 ); : : : ; λ(pk ) ; ν1 νk where n = p1 : : : pk is the prime factorization of n. Clearly, λ(1) = 1. Despite their many similarities, the functions '(n) and λ(n) often exhibit remarkable differences in their arithmetic behavior, and a vast number of results about the growth rate and various arithmetical properties of '(n) and λ(n) have been obtained; see for example [4, 5, 7, 8, 9, 11, 15]. -
Overpseudoprimes, Mersenne Numbers and Wieferich Primes 2
OVERPSEUDOPRIMES, MERSENNE NUMBERS AND WIEFERICH PRIMES VLADIMIR SHEVELEV Abstract. We introduce a new class of pseudoprimes - so-called “overpseu- doprimes” which is a special subclass of super-Poulet pseudoprimes. De- noting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2p − 1, where p is prime, and squares of Wieferich primes are overpseudoprimes. 1. Introduction n Sometimes the numbers Mn =2 − 1, n =1, 2,..., are called Mersenne numbers, although this name is usually reserved for numbers of the form p (1) Mp =2 − 1 where p is prime. In our paper we use the latter name. In this form numbers Mp at the first time were studied by Marin Mersenne (1588-1648) at least in 1644 (see in [1, p.9] and a large bibliography there). We start with the following simple observation. Let n be odd and h(n) denote the multiplicative order of 2 modulo n. arXiv:0806.3412v9 [math.NT] 15 Mar 2012 Theorem 1. Odd d> 1 is a divisor of Mp if and only if h(d)= p. Proof. If d > 1 is a divisor of 2p − 1, then h(d) divides prime p. But h(d) > 1. Thus, h(d)= p. The converse statement is evident. Remark 1. This observation for prime divisors of Mp belongs to Max Alek- seyev ( see his comment to sequence A122094 in [5]). -
Introduction to Abstract Algebra “Rings First”
Introduction to Abstract Algebra \Rings First" Bruno Benedetti University of Miami January 2020 Abstract The main purpose of these notes is to understand what Z; Q; R; C are, as well as their polynomial rings. Contents 0 Preliminaries 4 0.1 Injective and Surjective Functions..........................4 0.2 Natural numbers, induction, and Euclid's theorem.................6 0.3 The Euclidean Algorithm and Diophantine Equations............... 12 0.4 From Z and Q to R: The need for geometry..................... 18 0.5 Modular Arithmetics and Divisibility Criteria.................... 23 0.6 *Fermat's little theorem and decimal representation................ 28 0.7 Exercises........................................ 31 1 C-Rings, Fields and Domains 33 1.1 Invertible elements and Fields............................. 34 1.2 Zerodivisors and Domains............................... 36 1.3 Nilpotent elements and reduced C-rings....................... 39 1.4 *Gaussian Integers................................... 39 1.5 Exercises........................................ 41 2 Polynomials 43 2.1 Degree of a polynomial................................. 44 2.2 Euclidean division................................... 46 2.3 Complex conjugation.................................. 50 2.4 Symmetric Polynomials................................ 52 2.5 Exercises........................................ 56 3 Subrings, Homomorphisms, Ideals 57 3.1 Subrings......................................... 57 3.2 Homomorphisms.................................... 58 3.3 Ideals.........................................